Spectral and regularity properties of an operator calculus related to Landau quantization

TL;DR

This paper extends Weyl operator theory to Landau--Weyl pseudodifferential operators, enabling analysis of spectral and regularity properties of quantum systems in magnetic fields using modulation spaces.

Contribution

It introduces Landau--Weyl calculus and intertwining transforms, linking it to standard Weyl calculus, and applies it to spectral analysis and regularity of Schrödinger equations.

Findings

Eigenvalues and eigenfunctions of Landau Hamiltonian recovered easily.

Established global hypoellipticity results for related operators.

Studied regularity of solutions to Schrödinger equations in magnetic fields.

Abstract

The theme of this work is that the theory of charged particles in a uniform magnetic field can be generalized to a large class of operators if one uses an extended a class of Weyl operators which we call "Landau--Weyl pseudodifferential operators". The link between standard Weyl calculus and Landau--Weyl calculus is made explicit by the use of an infinite family of intertwining "windowed wavepacket transforms"; this makes possible the use of the theory of modulation spaces to study various regularity properties. Our techniques allow us not only to recover easily the eigenvalues and eigenfunctions of the Hamiltonian operator of a charged particle in a uniform magnetic field, but also to prove global hypoellipticity results and to study the regularity of the solutions to Schroedinger equations.